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Abstract:
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Given n samples of p-dimensional iid multivariate Gaussian, we are interested in testing a null hypothesis that the population covariance matrix Sigma is the identity against an alternative hypothesis where Sigma is a spike matrix; i.e., all eigenvalues are 1, except for r of them are larger than 1 (i.e., spiked eigenvalues). We propose CuSum and Higher Criticism as two new tests, and also investigate a trace-based test and the Tracy-Widom test.
We consider a Rare/Weak setting where the spikes are both sparse and individually weak (i.e., 1<<r<<p, and each spiked eigenvalue is only larger than $1$ by a small amount). We discover the following phase transition: the two-dimensional phase space calibrating the spike sparsity and strengths partitions into the Region of Impossibility and the Region of Possibility. In Region of Impossibility, all tests are (asymptotically) powerless in separating the alternative and the null. In Region of Possibility, both the CuSum test and the trace-based test have asymptotically full power. The TW test, however, could be powerless in a sub-region of the Region of Possibility.
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